How Do I Simplify the Function f(x) = cos(x)(3cos(3x)) + (sin(3x))(-sin(x))?

[Gill]

hey, i need to simplify this, and I am having a brain fart ...can u help me?

f(x)= cosx(3cos3x) + (sin3x)(-sinx)

i then have to find f(pi/6), but I am hoping i can do that once this is simplified...

thanks
Gill

 

[maverick6664]

hey, i need to simplify this, and I am having a brain fart ...can u help me?
f(x)= cosx(3cos3x) + (sin3x)(-sinx)
i then have to find f(pi/6), but I am hoping i can do that once this is simplified...
thanks
Gill

I wonder you made a typo... is cosx(3cos3x) $$\cos(x*3\cos(3x))$$ or $$\cos(x)*3\cos(3x)$$? Please place braces and it will clarify. But as for f(pi/6), it looks easy in any typo cases, because $$\cos(3*\pi/6) = 0, \sin(3*\pi/6) = 1$$

 

[Architeuthis Dux]

Sure, I can definitely help you simplify this expression. The first step would be to use the trigonometric identity cos(A+B) = cosAcosB - sinAsinB to rewrite the first term as cosx(cos3x) - sinx(sin3x). Then, we can use another identity, sin(A+B) = sinAcosB + cosAsinB, to rewrite the second term as (sin3x)(cosx) - (cos3x)(sinx).

Next, we can distribute the coefficients to get cosxcos3x - sinxsin3x + sin3xcosx - cos3xsinx. Now, we can use the fact that cosxcos3x = 1/2(cos(x+3x) + cos(x-3x)) and sinxsin3x = 1/2(cos(x-3x) - cos(x+3x)) to simplify further. This gives us 1/2(cos4x) + 1/2(cos2x) - 1/2(sin4x) + 1/2(sin2x).

Finally, we can use the double angle identities cos2x = 1/2(1 + cos4x) and sin2x = 1/2(1 - cos4x) to get our final simplified expression: 1/2 + 1/4(cos4x) - 1/4(sin4x).

Now, to find f(pi/6), we can simply substitute pi/6 for x in our simplified expression. I hope this helps and good luck with your calculations!